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1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME 136 CONSTRUCTING A NEW FAMILY DISTRIBUTION FROM THREE PARAMETERS WEIBULL USING ENTROPY TRANSFORMATION Dhwyia S. Hassun1 1 Professor, Department of Statistics, College of Administration & Economic, University of Baghdad, Iraq ABSTRACT This paper deals with constructing a new family probability density function from Weibull three parameters (ߠ, ߚ, ܿ), using statistical entropy transformation (which is considered as a measure of uncertainty), we construct this p.d.f, then prove its integrate from (ߠ ݐ ∞) equal (1). Also the cumulative distribution function (C.D.F) was derived, and given in simple form which is necessary for simulation procedure. The constructed . ݀. ݂ is necessary for skewed data, and for data represent time to failure of device and component after a value of threshold parameter. All the derivation required are explained. The formula of (ݎ௧ ) moments is derived in order to be used for estimation of parameters by moment method. The maximum likelihood estimators of three parameters were found, it is implicit functions, which need numerical solution to obtain (ߚመொ. ߠொ, ܿ̂ொ). Keywords: New Generated . ݀. ݂, Entropy Transformation, Maximum Likelihood & Moment Estimator, Cumulative Distribution Function. 1. INTRODUCTION One of the commonly used probability distribution for studying the reliability and maintainability analysis is the three parameters Weibull. The use of Weibull distribution to describe real phenomena has a long history. This distribution was originally proposed by the Swedish physicist WaloddiWeibull. He used it for modeling the distribution of breaking strength of materials. Since then it has received applications in many areas (Weibull 1951). This distribution has many applications in engineering, reliability, break age data, and time to failure data. It is applied in optimality testing of Markov type optimization[15] , also it is useful to study the voltage of electric circuits[16] . The estimating of three parameters Weibull distribution is complicated, since the equations obtained from the estimation method are too difficult to implement it, but in this paper we work on modifying the . ݀. ݂ of three parameters Weibull, using entropy like INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING AND TECHNOLOGY (IJARET) ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME: http://www.iaeme.com/IJARET.asp Journal Impact Factor (2014): 7.8273 (Calculated by GISI) www.jifactor.com IJARET © I A E M E
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME 137 transformation, this modified . ݀. ݂ is useful in cases of uncertainty found in data obtained from various applications, so the work here is restricted only about the new generated family with its derivation and how to estimate its three parameters using method of moments and maximum likelihood method. But in future we shall continue the work and apply simulation procedure to compare between various estimators of parameters. 2. OBJECTIVE OF RESEARCH The objective of this paper is to find a new generated probability distribution function using entropy like transformation of Weibull three parameters, also to derive its cumulative distribution function, also to derive a general form of moments from; ܧሺݔሻ ൌ ܧ ቈ൬ ݕ െ ߠ ߚ ൰ And use it to find first second and third moments aboutorigin. Therefore the generated probability density function from Weibull three parameters (c is shape parameter and ߠ location parameter and ߚ scale parameter), is derived and it is defined by; ݑሺݐሻ ൌ ൬ ܿ ߚ ൰ ൬ ݕ െ ߠ ߚ ൰ ଶିଵ exp ቊെ ൬ ݕ െ ߠ ߚ ൰ ቋ ݕ ߠ 0 ݓ⁄ Where; න ݑሺݐሻ݀ݐ ൌ 1 ∞ ఏ Also we explain how to estimate the parameters (ߠ, ߚ, ܿ), and how to find the cumulative distribution function of [ݑሺݐሻ] i.e [ݎሺܶ ݐሻ], ܨሺݐሻ ൌ න ൬ ܿ ߚ ൰ ൬ ݕ െ ߠ ߚ ൰ ଶିଵ exp ቊെ ൬ ݕ െ ߠ ߚ ൰ ቋ ݀ݕ ௧ ఏ ܨሺݐሻ ൌ 1 െ exp ቊെ ൬ ݐ െ ߠ ߚ ൰ ቋ ቈ൬ ݐ െ ߠ ߚ ൰ 1 3. DEFINITION OF CONSTRUCTED . . FUNCTION We know that the cumulative distribution function ܿ. ݀. ݂ of .ݎ ݒ Weibull with three parameters (c is shape parameter and ߠ location parameter and ߚ scale parameter) is defined by; ܨሺݐሻ ൌ 1 െ exp ቄെ ቀ ௧ିఏ ఉ ቁ ቅ ݐ ߠ (1) And the reliability function is; ܴሺݐሻ ൌ exp ቄെ ቀ ௧ିఏ ఉ ቁ ቅ (2)
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME 138 Then the generated function ݃ሺݐሻ defined as; ݃ሺݐሻ ൌ ܨሺݐሻ ܴሺݐሻ ln ܴሺݐሻ (3) ൌ 1 െ exp ቄെ ቀ ௧ିఏ ఉ ቁ ቅ െ ቀ ௧ିఏ ఉ ቁ exp ቄെ ቀ ௧ିఏ ఉ ቁ ቅ (4) ݃ሺݐሻdefined in equation (3) using entropy like transformation is derived w.r.to ()ݐ to obtain the new generated . ݀. ݂ ݑሺݐሻ; ݃′ሺݐሻ ൌ ݑሺݐሻ ൌ ቀ ఉ ቁ ቀ ௧ିఏ ఉ ቁ ଶିଵ exp ቄെ ቀ ௧ିఏ ఉ ቁ ቅ ݐ ߠ ߠ, ߚ, ܿ 0 (5) 0 ݓ⁄ Which integrate to one (i.e); ݑሺݐሻ݀ݐ ൌ 1 ∞ ఏ (6) Also the cumulative distribution function (C.D.F) is obtained; ܨ்ሺݐሻ ൌ ݎሺܶ ݐሻ ൌ ݑሺݕሻ݀ݕ ௧ ఏ (7) ൌ න ൬ ܿ ߚ ൰ ൬ ݕ െ ߠ ߚ ൰ ଶିଵ exp ቊെ ൬ ݕ െ ߠ ߚ ൰ ቋ ݀ݕ ∞ ఏ Using integration by parts we have; ܨሺݐሻ ൌ 1 െ exp ቄെ ቀ ௧ିఏ ఉ ቁ ቅ ቂቀ ௧ିఏ ఉ ቁ 1ቃ (8) In order to assess the performances of the parameter estimation methods by moments and M.L and other methods, first we derive a formula for (ݎ௧ ) moments of ቀ ௧ିఏ ఉ ቁi.eܧ ቀ ௧ିఏ ఉ ቁ which is defined in equation (10), were; ܧ ቀ ௧ିఏ ఉ ቁ ൌ ቀ ௧ିఏ ఉ ቁ ݂ሺݐሻ݀ݐ ∞ ఏ (9) ൌ ൬ ܿ ߚ ൰ න ൬ ݐ െ ߠ ߚ ൰ ଶାିଵ exp ቊെ ൬ ݐ െ ߠ ߚ ൰ ቋ ݀ݐ ∞ ఏ Let ݕ ൌ ௧ିఏ ఉ ݀ݕ ൌ ௗ௧ ఉ ݀ݐ ൌ ߚ݀ݕ ߠ ൏ ݕ ൏ ∞ ܧሺݕሻ ൌ ൬ ܿ ߚ ൰ න ݕଶାିଵ ݁ݔሼെݕሽ ߚ ݀ݕ ∞
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME 139 Let ݖ ൌ ݕ ݕ ൌ ݖ భ ݀ݕ ൌ ଵ ݖ భ ିଵ ܧሺݕሻ ൌ ܿ න ሺݖ భ ሻଶାିଵ exp ሺെݖሻ ൬ 1 ܿ ൰ ቀݖ భ ିଵ ቁ ݀ݖ ∞ ൌ න ሺݖሻଶା ି భ ା భ ିଵ ݁ି ݖ ݀ݖ ∞ ൌ න ሺݖሻଶା ିଵ exp ሺെݖሻ ݀ݖ ∞ ൌ Γ ቀ2 ݎ ܿ ቁ ܧሺݕሻ ൌ Γ ቀ2 ݎ ܿ ቁ ܧ ቀ ௧ିఏ ఉ ቁ ൌ Γ ቀ2 ቁ (10) When ݎ ൌ 1; ܧ ቀ ௧ ି ఏ ఉ ቁ ൌ Γ ቀ2 ଵ ቁ (11) ܧሺݐሻ ൌ ߚ Γ ൬2 1 ܿ ൰ ߠ And from; ܧ ൬ ݐ െ ߠ ߚ ൰ ଶ ൌ Γ ൬2 2 ܿ ൰ ܧሺݐ െ ߠሻଶ ൌ ߚଶ Γ ቀ2 ଶ ቁ (12) Also; ܧሺݐ െ ߠሻଷ ൌ ߚଷ Γ ቀ2 ଷ ቁ (13) These expectations are useful in obtaining moment estimators for the three parameters (ߠ, ߚ, ܿ): From ܧሺݐሻ ൌ ∑ ௧ ݐ ൌ ߚመΓ ቀ2 ଵ ̂ ቁ ߠ (14) And ܧሺݐଶሻ ൌ ∑ ௧ మ ܧሺݐଶሻ ൌ 2ߠܧሺݐሻ ߚଶ Γ ൬2 2 ܿ ൰ െ ߠଶ ∑ ݐ ଶ ݊ ൌ 2ߠ ൜ߚመΓ ൬2 1 ܿ ൰ ߠൠ ߚመଶ Γ ൬2 2 ܿ ൰ െ ߠ ଶ ∑ ݐ ଶ ݊ ൌ 2ߠߚመΓ ൬2 1 ܿ ൰ 2ߠଶ ߚመଶ Γ ൬2 2 ܿ ൰ െ ߠଶ
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME 140 ∑ ௧ మ ൌ 2ߠݐ ߚመଶ Γ ቀ2 ଶ ቁ െ ߠଶ (15) ∑ ݐ ݊ ൌ ߚመΓ ൬2 1 ܿ̂ ൰ ߠ And from third moment we have; ܧሺݐଷ െ 3ݐଶ ߠ 3ߠݐଶ െ ߠଷሻ ൌ ߚଷ Γ ൬2 3 ܿ ൰ ܧሺݐଷሻ ൌ ߚଷ Γ ൬2 3 ܿ ൰ 3ߠܧሺݐଶሻ െ 3ߠଶ ܧሺݐሻ ߠଷ ∑ ݐ ଷ ݊ ൌ ߚଷ Γ ൬2 3 ܿ ൰ 3ߠ ൜2ߠܧሺݐሻ ߚଶ Γ ൬2 2 ܿ ൰ െ ߠଶ ൠ െ 3ߠଶ ൜ߚΓ ൬2 1 ܿ ൰ ߠൠ ߠଷ ܧሺݐሻ ൌ ߚΓ ൬2 1 ܿ ൰ ߠ ∑ ݐ ଷ ݊ ൌ ߚଷ Γ ൬2 3 ܿ ൰ 6ߠଶ ܧሺݐሻ 3ߠߚଶ Γ ൬2 2 ܿ ൰ െ 3ߠଷ െ 3ߠଶ ߚΓ ൬2 1 ܿ ൰ െ 3ߠଷ ߠଷ ܧሺݐሻ ൌ ߚΓ ൬2 1 ܿ ൰ ߠ 6ߠଶ ܧሺݐሻ ൌ 6ߠଶ ߚΓ ൬2 1 ܿ ൰ 6ߠଷ ∑ ݐ ଷ ݊ ൌ ߚଷ Γ ൬2 3 ܿ ൰ 6ߠଶ ߚΓ ൬2 1 ܿ ൰ 6ߠଷ 3ߠߚଶ Γ ൬2 2 ܿ ൰ െ 6ߠଷ െ 3ߠଶ ߚΓ ൬2 1 ܿ ൰ ߠଷ ∑ ௧ య ൌ ߚଷ Γ ቀ2 ଷ ቁ 3ߠଶ ߚΓ ቀ2 ଵ ቁ 3ߠߚଶ Γ ቀ2 ଶ ቁ ߠଷ (16) There for to find moment estimators for the three parameters (ߠ, ߚ, ܿ) of new generated . ݀. ݂ the following three equations must solved numerically and simultaneously, these equations (14, 15, 16) are; ∑ ݐ ݊ ൌ ߚመΓ ൬2 1 ܿ̂ ൰ ߠ ∑ ݐ ଶ ݊ ൌ 2ߠݐ ߚመଶ Γ ൬2 2 ܿ ൰ െ ߠଶ ∑ ݐ ଷ ݊ ൌ ߚመଷ Γ ൬2 3 ܿ ൰ 3ߠଶ ߚመΓ ൬2 1 ܿ ൰ 3ߠߚመଶ Γ ൬2 2 ܿ ൰ ߠଷ 4. ESTIMATING THREE PARAMETERS BY MAXIMUM LIKELIHOOD METHOD To find M.L.E for three parameters (ߠ, ߚ, ܿ) of the generated . ݀. ݂ (equation [5]), let (ݐଵ, ݐଶ, … , ݐ) be value of a random variable (T) with (݊) sample size, from (5) then; ܮ ൌ ෑ ൬ ܿ ߚ ൰ ൬ ݐ െ ߠ ߚ ൰ ଶିଵ ݁ݔ ቊെ ൬ ݐ െ ߠ ߚ ൰ ቋ ୀଵ
6.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME 141 ܮ ൌ ൬ ܿ ߚ ൰ ෑ ൬ ݐ െ ߠ ߚ ൰ ଶିଵ ݁ݔ ൝െ ൬ ݐ െ ߠ ߚ ൰ ୀଵ ൡ ୀଵ ݐ ߠ ሺ17ሻ Taking logarithm of (17) yield; log ܮ ൌ ݊ log ܿ െ ݊ log ߚ ሺ2ܿ െ 1ሻ log ൬ ݐ െ ߠ ߚ ൰ െ ൬ ݐ െ ߠ ߚ ൰ ୀଵ ୀଵ ሺ18ሻ Simplifying equation (18) implies; log ܮ ൌ ݊ log ܿ െ ݊ log ߚ ሺ2ܿ െ 1ሻ logሺݐ െ ߠሻ െ ݊ሺ2ܿ െ 1ሻ log ߚ െ ൬ ݐ െ ߠ ߚ ൰ ୀଵ ୀଵ ሺ19ሻ Then differentiating (19) partially with respect to each parameter we obtain MLE from equating each partial derivatives to zero. ߲ log ܮ ߲ܿ ൌ ݊ ܿ 2 logሺݐ െ ߠሻ െ 2݊ log ߚ െ ൬ ݐ െ ߠ ߚ ൰ log ൬ ݐ െ ߠ ߚ ൰ ୀଵ ୀଵ ሺ20ሻ ߲ log ܮ ߲ߚ ൌ െ ݊ ߚ െ ݊ሺ2ܿ െ 1ሻ ߚ ܿߚିିଵ ሺݐ െ ߠሻ ୀଵ ሺ21ሻ ߲ log ܮ ߲ߠ ൌ ሺ2ܿ െ 1ሻ ൬ 1 ݐ െ ߠ ൰ ሺെ1ሻ െ ܿߚି ሺݐ െ ߠሻିଵ ୀଵ ୀଵ ሺെ1ሻ ฺ െሺ2ܿ െ 1ሻ ሺݐ െ ߠሻିଵ ୀଵ ܿߚି ሺݐ െ ߠሻିଵ ୀଵ ൌ 0 ሺݐ െ ߠሻିଵ ܿߚି ሺݐ െ ߠሻ െ ሺ2ܿ െ 1ሻ ୀଵ ൩ ൌ 0 ୀଵ ֜ ܿߚି ሺݐ െ ߠሻ െ ሺ2ܿ െ 1ሻ ୀଵ ൌ 0 ሺ22ሻ Then the equations for maximum likelihood estimators of three parameters which are obtained from (20, 21, 22) are; 2 logሺݐ െ ߠሻ െ ൬ ݐ െ ߠ ߚ ൰ log ൬ ݐ െ ߠ ߚ ൰ ୀଵ ൌ ݊ሺ2 log ߚ െ 1 ܿ ሻ ୀଵ
7.
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME 142 ܿ̂ ൫ݐ െ ߠ൯ ̂ ൌ ቆ 2݊ܿ̂ ሺߚመሻ̂ ቇ ୀଵ ൫ݐ െ ߠ൯ ̂ ൌ ቆ 2݊ ሺߚመሻ̂ ቇ ୀଵ Therefore; ߚመொ ൌ ቈ ∑ ൫௧ିఏಾಽಶ൯ ො సభ ଶ భ ො (23) Which is implicit function of ߠொand (ܿ̂ொ). Also we have; ܿ̂ ቆ ݐ െ ߠ ߚመ ቇ ̂ ൌ 2 ܿ̂ 1 ୀଵ 2 1 ܿ̂ ൌ ቆ ݐ െ ߠ ߚመ ቇ ̂ ୀଵ 1 ܿ̂ொ ൌ ቆ ݐ െ ߠ ߚመ ቇ ̂ െ 2 ୀଵ ܿ̂ொ ൌ ቆ ݐ െ ߠ ߚመ ቇ ̂ െ 2 ୀଵ ൩ ିଵ ሺ24ሻ Which is also an implicit function can be solved numerically, and from డ ୪୭ డఏ ൌ 0 ሺ2ܿ̂ െ 1ሻ ൫ݐ െ ߠ൯ ିଵ ൌ ܿ̂ሺߚመሻି̂ ୀଵ ൫ݐ െ ߠ൯ ̂ିଵ ୀଵ ሺ2ܿ̂ െ 1ሻ ൌ ܿ̂ሺߚመሻି̂ ൫ݐ െ ߠ൯ ̂ ୀଵ ቆ ݐ െ ߠ ߚመ ቇ ̂ ൌ ൬ 2ܿ̂ െ 1 ܿ̂ ൰ ୀଵ ቆ ݐ െ ߠ ߚመ ቇ ̂ ൌ ൬2 െ 1 ܿ̂ ൰ ୀଵ ሺ25ሻ Which is the same as equation (24), then for fixed value of (ߠ) equation (24 & 25) can be solved numerically to find (ߚመொ) and (ߠொ) and then (ܿ̂ொ).
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International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME 143 CONCLUSION 1. The new generated . ݀. ݂ using entropy transformation is useful for skewness data. 2. The data have location parameter so this distribution is good for representing the distribution of failure time after (ߠ) value since (ݐ ߠ). 3. The value of (ߠ) may represent the scale for evaluating the reliability of device after time interval since (ݐ ߠ). 4. Another methods for estimation can be used like Baysian method, Percentile method. REFERENCES [1] AlizadehNoughabi, H. (2010), A new estimator of entropy and its application in testing normality. J. Statist. Comput.Simul., 80,1151-1162. [2] AlizadehNoughabi, H. and Arghami, N. R. (2011a), Monte Carlo comparison of five exponentiality tests using different entropy estimates.J. Statist. Comput.Simul., 81, 1579-1592. [3] Bartkute V. and Sakalauskas L. (2007), Three parameter estimation of the Weibull distribution by order statistics; In: Recent advances in stochastic modeling and data analysis, Chania, Greece 29 May–1 June, 2007. New Jersey . . . [etc.], World Scientific, pp. 91–100. [4] H. Samimi, M. Akbari, (2013), "An Accurate Approximation to 3-Parameter Weibull Sum Distribution", International Research Journal of Applied and Basic Sciences, Vol, 4 (6): 1524-1529 . [5] Koutrouvelis, I. A., Canavos, G. C., and Meintanis, S. G. (2005). Estimation in the three parameter inverse Gaussian distribution. Computational Statistics & Data Analysis, 49, 1132-1147. [6] Kundu, D. and Raqab, M. Z. (2009). Estimation of R = P(Y < X) for three- parameter Weibull distribution. Statistics and Probability Letters 79, 1839- 1846. [7] Mahdi Teimouri1, and Arjun K. Gupta, (2013), "On the Three-Parameter Weibull Distribution Shape Parameter Estimation", Journal of Data Science 11, 403-414. [8] MazenZaindin, Ammar M. Sarhan, (2009), "Parameters Estimation of the Modified Weibull Distribution", Applied Mathematical Sciences, Vol. 3, 2009, no. 11, 541 - 550. [9] MiljenkoMarusicDarijaMarkovi and Dragan Juki, (2010), "Least squares fitting the three- parameter inverse Weibull density", Math. Commun., Vol. 15, No. 2, pp. 539-553. [10] Murthy, D. N. P., Xie, M. and Jiang, R. (2004). Weibull Models. John Wiley, New York. [11] Robert M. Gray, (2013), "Entropy and Information Theory", First Edition, Corrected, Springer- Verlag New York. [12] Sanjay Kumar Singh, Umesh Singh, and Dinesh Kumar.(2013), "Bayesian estimation of parameters of inverse Weibull distribution". Journal of Applied Statistics, 40(7):1597–1607. [13] VaidaBartkut˙e and Leonidas Sakalauskas, (2008), "The method of three-parameter Weibull distribution estimation", ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 12. [14] Vexler, A. and Gurevich, G. (2010), Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy. Comput.Statist.Data Anal., 54, 531-545. [15] Zilinskas A. and Zhigljavsky A.A. (1991), "Methods of the global extreme searching", Nauka, Moscow, (Russian). [16] Zhigljavsky A.A. (1985), "Mathematical theory of the global random search St.", Petersburg University Press, (Russian). [17] Weibull W.(1951), "A statistical distribution functions with wide applicability", J.Appl. Mech. 18, 293-297. [18] Faris M. Al-Athari, “Moment Properties of Two Maximum Likelihood Estimators of the Mean of Truncated Exponential Distribution”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 7, 2013, pp. 258 - 265, ISSN Print: 0976-6480, ISSN Online: 0976-6499.
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